We investigate the interactions between scalar implicatures and presuppositions
in sentences containing both a scalar item and presupposition trigger We first
critically discuss Gajewski and Sharvit’s previous approach We then closely examine
two ways of integrating an exhaustivity - based theory of scalar implicatures with a
trivalent approach to presuppositions The empirical side of our discussion focuses on
two novel observations (i) the interactions between prosody and monotonicity and
(ii) what we call presupposed ignorance In order to account for these observations
our final proposal relies on two mechanisms of scalar strengthening the Presupposed
Ignorance Principle and an exhaustivity operator which lets the presuppositions of
negated alternatives project.

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• Presupposed ignorance and exhaustification: how scalar implicatures and presuppositions interact